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Full text of "Analytical Mechanics"

resultant torque equals and is oppositely directed to the angular kinetic reaction.
Resultant torque = (angular kinetic reaction).    (I)
In order to understand the nature of the angular kinetic reaction .consider the following experiment: If we try to rotate a flywheel, which is free to move about a horizontal axis, by pulling at one end of a string which is wound around the axle, Fig, 110, we find that the greater the angular velocity which we want to impart in a given interval of time the harder we must pull at the string. But since the pull
of the string and the reaction of the bearings form a couple and since the increase in the angular velocity per unit time means angular acceleration, we conclude that a torque must be applied to the flywheel in order to impart to it an angular acceleration, and that the greater the acceleration desired the greater must the torque be. Evidently the torque which we apply to the flywheel expends itself in overcoming certain reactions. The resisting torque due to the friction between the axle and its bearings and between the surface of the flywheel and the surrounding air must be overcome. But if we gradually diminish this resisting torque by reducing the friction we observe that the torque which must be applied, in order to give the flywheel a certain angular acceleration, tends towards a constant value different from zero. In other words even if all the resisting torques due to friction were eliminated we would have to apply a torque of definite magnitude in order to give the flywheel a desired angular acceleration; that is, the flywheel resists torques which impart to it an angular acceleration. This resistance to angular acceleration is the angular kinetic reaction.